ALG-00 · laboratory
Quadratic formula, AM–GM, De Moivre, logarithms and the doubling time.
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26 governing sheets
Open the drawing benchx = (−b ± √(b² − 4ac)) / (2a) for ax² + bx + c = 0.
Algebra(x+y)/2 ≥ √(xy) for x,y ≥ 0, equality iff x = y.
Inequalities(x+y)/2 ≥ 2xy/(x+y) for x,y > 0. AM ≥ HM.
Inequalities[r (cos θ + i sin θ)]^n = r^n (cos nθ + i sin nθ).
Complexlog_b a = ln a / ln b.
Logarithmsy = a^x = e^{x ln a} for a > 0.
Exponentialax² + bx + c = a(x + b/2a)² + (c − b²/4a). Vertex form.
Algebra(a+b)² = a² + 2ab + b².
AlgebraP(r) is the remainder of P(x) divided by (x − r). Quadratic snapshot.
Polynomials(x − r) divides P iff P(r) = 0. The sheet reports P(r) as a zero-test.
PolynomialsS = a (1 − r^n) / (1 − r) for r ≠ 1. n terms, first term a.
SeriesS = n/2 (2a + (n−1)d). n terms, first a, common difference d.
SeriesT2 = ln 2 / k for y = y0 e^{kt}. Time to double.
ExponentialA = P (1 + r/n)^{nt}. Principal P after t years.
Exponentiallog(xy) = log x + log y. The sheet checks both sides in ln.
Logarithms|z| = √(x² + y²) for z = x + iy.
Complexz z̄ = x² + y² = |z|². Conjugation flips the sign of y.
ComplexΔ = b² − 4ac. Sign of Δ classifies the roots.
AlgebraFor ax² + bx + c, x1 + x2 = −b/a. For a cubic, x1+x2+x3 = −b/a.
Polynomialsy = y0 e^{kt}. Continuous growth or decay.
Exponentialln(x/y) = ln x − ln y.
LogarithmsH = 2xy / (x+y). Reciprocal of the mean of reciprocals.
MeansFinite geometric.
AlgebraExponential growth.
AlgebraPolar radius.
AlgebraAny base.
Algebra